Methods for compression using a denoiser

ABSTRACT

Various embodiments of the present invention provide a compression method and system that compresses received data by first denoising the data and then losslessly compressing the denoised data. Denoising removes high entropy features of the data to produce lower entropy, denoised data that can be efficiently compressed by a lossless compression technique. One embodiment of the invention is a universal lossy compression method obtained by cascading a denoising technique with a universal lossless compression method. Alternative embodiments include methods obtained by cascading a denoising technique with one or more lossy or lossless compression methods.

TECHNICAL FIELD

The present invention is related to data compression methods and systemsand, in particular, to a compression method that employs both a denoiserand a compression method.

BACKGROUND OF THE INVENTION

Compression techniques are widely used in computing, data storage, andelectronic communications for decreasing the volume of data to allowmore efficient storage and/or transmission. For example, currentmodem-based and DSL interconnections do not provide sufficient datatransmission bandwidth to allow for transmission of uncompressed,real-time video signals at resolutions close to the resolution ofbroadcast television. However, highly compressed video signals can betransmitted through such internet connections and decompressed anddisplayed on a user's computer.

Data compression can be carried out in either a lossy or a losslessfashion. Lossy compression can generally provide much better compressionratios, but the decompressed data is generally distorted with respect tothe original data. For example, in lossy compression of video signals,the decompressed video signal may have lower resolution and lowerdisplayed frame rates. By contrast, lossless compression compresses dataso that the compressed data can be accurately decompressed toidentically restore the original data.

Many lossless and lossy compression methods assume a statistical modelof the data being compressed. Discrepancies between the model assumed bya method and the actual statistical properties of the data result inpoor compression and/or increased distortion in the reconstructedsignal. Universal compression methods mitigate this problem by adaptingmethod parameters to better suit the actual data being compressed. Thevarious embodiments of the Lempel-Ziv compression method comprise oneset of successful and widely deployed universal lossless compressionmethods. The Lempel-Ziv method and other lossless universal compressionmethods also possess certain optimality properties in a variety offormal mathematical settings involving data generated by stochasticprocesses and/or classes of competing data-tuned compression methods. Incontrast to the lossless case, all universal lossy compression methodsknown to be optimal in the formal mathematical settings are hopelesslycomplex computationally and hence impractical. Practical universal lossycompression methods are therefore heuristically driven. Designers,developers, and users of compression methods and systems are constantlyseeking new compression techniques that provide better computationalefficiency and other advantages, such as improved heuristics foruniversal lossy compression methods.

SUMMARY OF THE INVENTION

Various embodiments of the present invention provide a lossy compressionmethod and system that compresses received data by first denoising thedata and then compressing the denoised data. Denoising removes highentropy features of the data to produce lower entropy, denoised datathat can be efficiently compressed by a lossless compression technique.One embodiment of the invention is a universal lossy compression methodobtained by cascading a denoising technique with a universal losslesscompression method. Alternative embodiments include methods obtained bycascading a denoising technique with one or more lossy or losslesscompression methods.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 illustrates introduction of noise into a clean signal to producea noisy signal and subsequent denoising of the noisy signal to produce arecovered signal.

FIGS. 2A-D illustrate a motivation for a discrete denoiser related tocharacteristics of the noise-introducing channel.

FIGS. 3A-D illustrate a context-based, sliding window approach by whicha discrete denoiser characterizes the occurrences of symbols in a noisysignal.

FIG. 4 illustrates a convenient mathematical notation and data structurerepresenting a portion of the metasymbol table constructed by a discretedenoiser, as described with reference to FIGS. 3A-D.

FIGS. 5A-D illustrate the concept of symbol-corruption-relateddistortion in a noisy or recovered signal.

FIG. 6 displays one form of the symbol-transformation distortion matrixΛ.

FIG. 7 illustrates computation of the relative distortion expected fromreplacing a symbol “a_(a)” in a received, noisy signal by the symbol“a_(x.)”

FIG. 8 illustrates use of the column vector λ_(a) _(x) □π_(a) _(a) tocompute a distortion expected for replacing the center symbol a_(a) inthe metasymbol ba_(a)c in a noisy signal “s_(noisy)” by the replacementsymbol a_(x).

FIG. 9 shows estimation of the counts of the occurrences of symbols“a₁”-“a_(n)” for the clean signal.

FIG. 10 illustrates the process by which a discrete denoiser denoises anoisy, received signal.

FIGS. 11A-B illustrate the Lempel-Ziv lossless data compression.

FIG. 12 illustrates an encoding scheme for the vector quantization lossycompression technique.

FIG. 13 illustrates lossy compression by the vector quantizationtechnique described with reference to FIG. 12, using the encoding tableillustrated in FIG. 12.

FIG. 14 is a control-flow diagram for one general embodiment of thepresent invention.

FIG. 15 illustrates a self-tuning implementation of the losslesscompression method of various embodiments of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the present invention relate to compression techniques.To facilitate discussion of these embodiments of the present invention,a discrete denoiser implementation is discussed, in a first subsectionbelow, followed by discussion of lossless and lossy compressiontechniques, in a second subsection. Finally, in a third subsection,methods for compression using a denoiser are discussed.

Dude

FIG. 1 illustrates introduction of noise into a clean signal to producea noisy signal and subsequent denoising of the noisy signal to produce arecovered signal. In FIG. 1, signals are represented as sequences ofsymbols that are each members of an alphabet A having n distinctsymbols, where A is:A=(a ₁ , a ₂ , a ₃ , . . . a _(n))Note that the subscripts refer to the positions of the respectivesymbols within an ordered listing of the different symbols of thealphabet, and not to the positions of symbols in a signal. In FIG. 1, aninitial, clean signal 102 comprises an ordered sequence of nine symbolsfrom the alphabet A. In normal circumstances, an input signal wouldgenerally have thousands, millions, or more symbols. The short inputsignal 102 is used for illustrative convenience.

The clean signal 102 is transmitted or passed through anoise-introducing channel 104, producing a noisy signal 106. In theexample shown in FIG. 1, the output signal 106 comprises symbols fromthe same alphabet as the input signal 102, although, in general, theinput symbols may be chosen from a different, equally sized or smalleralphabet than that from which the output symbols are selected. In theexample shown in FIG. 1, the sixth symbol in the clean signal 108, “a₉,”is altered by the noise-introducing channel to produce the symbol “a₂”110 in the noisy signal 106. There are many different types ofnoise-introducing channels, each type characterized by the types andmagnitudes of noise that the noise-introducing channel introduces into aclean signal. Examples of noise-introducing channels include electroniccommunications media, data storage devices to which information istransferred and from which information is extracted, and transmissionand reception of radio and television signals. In this discussion, asignal is treated as a linear, ordered sequence of symbols, such as astream of alphanumeric characters that comprise a text file, but theactual data into which noise is introduced by noise-introducing channelsin real world situations may include two-dimensional images, audiosignals, video signals, and other types of displayed and broadcastinformation.

In order to display, broadcast, or store a received, noisy signal withreasonable fidelity with respect to the initially transmitted cleansignal, a denoising process may be undertaken to remove noise introducedinto the clean signal by a noise-introducing channel. In FIG. 1, thenoisy signal 106 is passed through, or processed by, a denoiser 112 toproduce a recovered signal 114 which, when the denoising process iseffective, is substantially closer to, or more perceptually similar to,the originally transmitted clean signal than to the received noisysignal.

Many types of denoisers have been proposed, studied, and implemented.Some involve application of continuous mathematics, some involvedetailed knowledge of the statistical properties of the originallytransmitted clean signal, and some rely on detailed informationconcerning time and sequence-dependent behavior of the noise-introducingchannel. The following discussion describes a discrete denoiser,referred to as “DUDE,” related to the present invention. The DUDE isdiscrete in the sense that the DUDE processes signals comprisingdiscrete symbols using a discrete algorithm, rather than continuousmathematics. The DUDE is universal in that it asymptotically approachesthe performance of an optimum denoiser employing knowledge of theclean-signal symbol-occurrence distributions.

The DUDE implementation is motivated by a particularnoise-introducing-channel model and a number of assumptions. These arediscussed below. However, DUDE may effectively function when the modeland assumptions do not, in fact, correspond to the particularcharacteristics and nature of a noise-introducing channel. Thus, themodel and assumptions motivate the DUDE approach, but the DUDE has amuch greater range of effectiveness and applicability than merely todenoising signals corrupted by a noise-introducing channel correspondingto the motivating model and assumptions.

As shown in FIG. 1, the DUDE 112 employs a particular strategy fordenoising a noisy signal. The DUDE considers each symbol within acontext generally comprising one or more symbols preceding and followingthe symbol according to a left to right ordering. For example, in FIG.1, the two occurrences of the symbol “a₂” in the noisy signal 106 occurwithin the same single preceding-and-following-symbol context. The fullcontext for the two occurrences of the symbol “a₂” in the noisy signal106 of the example in FIG. 1 is [“a₃,” “a₁”]. The DUDE either leaves allsymbols of a particular type “a_(i)” within a particular contextunchanged, or changes all occurrences of a particular type of symbol“a_(i)” within a particular context to a different symbol “a_(j).” Forexample, in FIG. 1, the denoiser has replaced all occurrences of thesymbol “a₂” 110 and 112 in the noisy signal within the full context[“a₃,” “a₁”] with the symbol “a₉” 114 and 116 in the recovered symbol.Thus, the DUDE does not necessarily produce a recovered signal identicalto the originally transmitted clean signal, but instead produces adenoised, recovered signal with less distortion with respect to theclean signal than the distortion within the noisy signal. In the aboveexample, replacement of the second symbol “a₂” 110 with the symbol “a₉”114 restores the originally transmitted symbol at that position, butreplacement of the first occurrence of symbol “a₂” 112 in the noisysignal with the symbol “a₉” 116 introduces a new distortion. The DUDEonly replaces one symbol with another to produce the recovered signalwhen the DUDE estimates that the overall distortion of the recoveredsignal with respect to the clean signal will be less than the distortionof the noisy signal with respect to the clean signal.

FIGS. 2A-D illustrate a motivation for DUDE related to characteristicsof the noise-introducing channel. DUDE assumes a memory-less channel. Inother words, as shown in FIG. 2A, the noise-introducing channel 202 maybe considered to act as a one-symbol window, or aperture, through whicha clean signal 204 passes. The noise-introducing channel 202 corrupts agiven clean-signal symbol, replacing the given symbol with anothersymbol in the nosy signal, with an estimateable probability that dependsneither on the history of symbols preceding the symbol through thenoise-introducing channel nor on the symbols that are subsequentlytransmitted through the noise-introducing channel.

FIG. 2B shows a portion of a table 206 that stores the probabilitiesthat any particular symbol from the alphabet A, “a_(i),” may becorrupted to a symbol “a_(j)” during transmission through thenoise-introducing channel. For example, in FIG. 2A, the symbol “a₆” 208is currently passing through the noise-introducing channel. Row 210 intable 206 contains the probabilities that symbol “a₆” will be corruptedto each of the different, possible symbols in the alphabet A. Forexample, the probability that the symbol “a₆” will be changed to thesymbol “a₁” 212 appears in the first cell of row 210 in table 206,indexed by the integers “6” and “1” corresponding to the positions ofsymbols “a₆” and “a₁” in the alphabet A. The probability that symbol“a₆” will be faithfully transferred, without corruption, through thenoise-introducing channel 214 appears in the table cell with indices (6,6), the probability of symbol “a₆” being transmitted as the symbol “a₆.”Note that the sum of the probabilities in each row of the table 206 is1.0, since a given symbol will be transmitted by the noise-introducingchannel either faithfully or it will be corrupted to some other symbolin alphabet A. As shown in FIG. 2C, table 206 in FIG. 2B can bealternatively expressed as a two-dimensional matrix Π 216, with thematrix element identified by indices (i, j) indicating the probabilitythat symbol “a_(i)” will be transmitted by the noise-introducing channelas symbol “a_(j).” Note also that a column j in matrix Π may be referredto as “π_(j)” or π_(a) _(j) .

As shown in FIG. 2D, a row vector 218 containing the counts of thenumber of each type of symbol in the clean signal, where, for example,the number of occurrences of the symbol “a₅” in the clean signal appearsin the row vector as m^(clean)[a₅], can be multiplied by thesymbol-transition-probability matrix Π 220 to produce a row vector 222containing the expected counts for each of the symbols in the noisysignal. The actual occurrence counts of symbols “a_(i)” in the noisysignal appear in the row vector m^(noisy). The matrix multiplication isshown in expanded form 224 below the matrix multiplication in FIG. 2D.Thus, in vector notation:m^(clean)Π≅m^(noisy)where

-   -   m^(clean) is a row vector containing the occurrence counts of        each symbol a_(i) in alphabet A in the clean signal; and    -   m^(noisy) is a row vector containing the occurrence counts of        each symbol a_(i) in alphabet A in the noisy signal.        The approximation symbol ≅ is employed in the above equation,        because the probabilities in the matrix Π give only the expected        frequency of a particular symbol substitution, while the actual        symbol substitution effected by the noise-introducing channel is        random. In other words, the noise-introducing channel behaves        randomly, rather than deterministically, and thus may produce        different results each time a particular clean signal is        transmitted through the noise-introducing channel. The error in        the approximation, obtained as the sum of the absolute values of        the components of the difference between the left and right        sides of the approximation, above, is generally small relative        to the sequence length, on the order of the square root of the        sequence length. Multiplying, from the right, both sides of the        above equation by the inverse of matrix Π, assuming that Π is        invertible, allows for calculation of an estimated row-vector        count of the symbols in the clean signal, {circumflex over        (m)}^(clean), from the counts of the symbols in the noisy        signal, as follows:        {circumflex over (m)}^(clean)=m^(noisy)Π⁻¹        In the case where the noisy symbol alphabet is larger than the        clean symbol alphabet, it is assumed that Π is full-row-rank and        the inverse in the above expression can be replaced by a        generalized inverse, such as the Moore-Penrose generalized        inverse.

As will be described below, the DUDE applies clean symbol countestimation on a per-context basis to obtain estimated counts of cleansymbols occurring in particular noisy symbol contexts. The actualdenoising of a noisy symbol is then determined from the noisy symbol'svalue, the resulting estimated context-dependent clean symbol counts,and a loss or distortion measure, in a manner described below.

As discussed above, the DUDE considers each symbol in a noisy signalwithin a context. The context may be, in a linear signal, such as thatused for the example of FIG. 1, the values of a number of symbolspreceding, following, or both preceding and following a currentlyconsidered signal. In 2-dimensional or higher dimensional signals, thecontext may be values of symbols in any of an almost limitless number ofdifferent types of neighborhoods surrounding a particular symbol. Forexample, in a 2-dimensional image, the context may be the eight pixelvalues surrounding a particular, interior pixel. In the followingdiscussion, a linear, 1-dimensional signal is used for examples, buthigher dimensional signals can be effectively denoised by the DUDE.

In order to consider occurrences of symbols within contexts in the1-dimensional-signal case, the DUDE needs to consider a number ofsymbols adjacent to each, considered symbol. FIGS. 3A-D illustrate acontext-based, sliding window approach by which the DUDE characterizesthe occurrences of symbols in a noisy signal. FIGS. 3A-D all employ thesame illustration conventions, which are described only for FIG. 3A, inthe interest of brevity. In FIG. 3A, a noisy signal 302 is analyzed byDUDE in order to determine the occurrence counts of particular symbolswithin particular contexts within the noisy signal. The DUDE employs aconstant k to describe the length of a sequence of symbols preceding,and the length of a sequence of symbols subsequent to, a particularsymbol that, together with the particular symbol, may be viewed as ametasymbol of length 2k+1. In the example of FIGS. 3A-D, k has the value“2.” Thus, a symbol preceded by a pair of symbols and succeeded by apair of symbols can be viewed as a five-symbol metasymbol. In FIG. 3A,the symbol “a₆” 304 occurs within a context of the succeeding k-lengthsymbol string “a₉a₂” 306 and is preceded by the two-symbol string “a₁a₃”308. The symbol “a₆” therefore occurs at least once in the noisy signalwithin the context [“a₁a₃,” “a₉a₂”], or, in other words, the metasymbol“a₁a₃a₆a₉a₂” occurs at least once in the noisy signal. The occurrence ofthis metasymbol within the noisy signal 302 is listed within a table 310as the first five-symbol metacharacter 312.

As shown in FIG. 3B, DUDE then slides the window of length 2k+1rightward, by one symbol, to consider a second metasymbol 314 of length2k+1. In this second metasymbol, the symbol “a₉” appears within thecontext [“a₃a₆,” “a₂a₁₇”]. This second metasymbol is entered into table310 as the second entry 316. FIG. 3C shows detection of a thirdmetasymbol 318 in the noisy signal 302 and entry of the third metasymbolinto table 310 as entry 320. FIG. 3D shows the table 310 followingcomplete analysis of the short noisy signal 302 by DUDE. Although, inthe examples shown in FIG. 3-D, DUDE lists each metasymbol as a separateentry in the table, in a more efficient implementation, DUDE enters eachdetected metasymbol only once in an index table, and increments anoccurrence count each time the metasymbol is subsequently detected. Inthis fashion, in a first pass, DUDE tabulates the frequency ofoccurrence of metasymbols within the noisy signal or, vieweddifferently, DUDE tabulates the occurrence frequency of symbols withincontexts comprising k preceding and k subsequent symbols surroundingeach symbol.

FIG. 4 illustrates a convenient mathematical notation and data structurerepresenting a portion of the metasymbol table constructed by DUDE, asdescribed with reference to FIGS. 3A-D. The column vectorm(s_(noisy),b,c) 402 represents a count of the occurrences of eachsymbol in the alphabet A within a particular context, represented by thek-length symbol vectors b and c, within the noisy signal s_(noisy),where the noisy signal is viewed as a vector. In FIG. 4, for example,the context value for which the occurrence counts are tabulated incolumn vector m(s_(noisy),b,c) comprises the symbol vector 404 and thesymbol vector 406, where k has the value 3. In the noisy signals_(noisy) 408, the symbol “a₃” 410 occurs within the context comprisingthree symbols 412 to the left of the symbol “a₃” 410 and three symbols414 to the right of the symbol “a₃”. This particular context has a valueequal to the combined values of symbol vectors 404 and 406, denoted[“a₇a₃a₆,” “a₅a₅a₅”] and this occurrence of the symbol “a₃” 410 withinthe context [“a₇a₃a₆,” “a₅a₅a₅”], along with all other occurrences ofthe symbol “a₃” in the context [“a₇a₃a₆,” “a₅a₅a₅”], is noted by a count416 within the column vector m(s_(noisy),b,c), with [b,c]=[“a₇a₃a₆,”“a₅a₅a₅”]. In other words, a symbol “a₃” occurs within the context[“a₇a₃a₆,” “a₅a₅a₅”] in the noisy signal s_(noisy) 321 times. The countsfor the occurrences of all other symbols “a₁”, “a₂”, and “a₄”-“a_(n)” inthe context [“a₇a₃a₆,” “a₅a₅a₅”] within noisy signal s_(noisy) arerecorded in successive elements of the column vector m(s_(noisy),“a₇a₃a₆”, “a₅a₅a₅”). An individual count within a column vectorm(s_(noisy),b,c) can be referred to using an array-like notation. Forexample, the count of the number of times that the symbol “a₃” appearsin the context [“a₇a₃a₆,” “a₅a₅a₅”] within the noisy signal s_(noisy),321, can be referred to as m(s_(noisy), “a₇a₃a₆”, “a₅a₅a₅”)[a₃].

DUDE employs either a full or a partial set of column vectors for alldetected contexts of a fixed length 2k in the noisy signal in order todenoise the noisy signal. Note that an initial set of symbols at thebeginning and end of the noisy signal of length k are not counted in anycolumn vector m(s_(noisy),b,c) because they lack either sufficientpreceding or subsequent symbols to form a metasymbol of length 2k+1.However, as the length of the noisy signal for practical problems tendsto be quite large, and the context length k tends to be relativelysmall, DUDE's failure to consider the first and final k symbols withrespect to their occurrence within contexts makes almost no practicaldifferent in the outcome of the denoising operation.

FIGS. 5A-D illustrate the concept of symbol-corruption-relateddistortion in a noisy or recovered signal. The example of FIGS. 5A-Drelates to a 256-value gray scale image of a letter. In FIG. 5A, thegray-scale values for cells, or pixels, within a two-dimensional image502 are shown, with the character portions of the symbol generallyhaving a maximum gray-scale value of 255 and the background pixelshaving a minimum gray-scale value of zero. Visual display of the imagerepresented by the two-dimensional gray-scale signal in FIG. 5A is shownin FIG. 5B 504. The gray-scale data in FIG. 5A is meant to represent alow resolution image of the letter “P.” As shown in FIG. 5B, the imageof the letter “P” is reasonably distinct, with reasonably high contrast.

FIG. 5C shows the gray-scale data with noise introduced by transmissionthrough a hypothetical noise-introducing channel. Comparison of FIG. 5Cto FIG. 5A shows that there is marked difference between the gray-scalevalues certain cells, such as cell 506, prior to, and after,transmission. FIG. 5D shows a display of the gray-scale data shown inFIG. 5C. The displayed image is no longer recognizable as the letter“P.” In particular, two cells contribute greatly to the distortion ofthe figure: (1) cell 506, changed in transmission from the gray-scalevalue “0” to the gray-scale value “223”; and (2) cell 508, changed intransmission from the gray-scale value “255” to the gray-scale value“10.” Other noise, such as the relatively small magnitude gray-scalechanges of cells 510 and 512, introduce relatively little distortion,and, by themselves, would have not seriously impacted recognition of theletter “P.” In this case, the distortion of the displayed imagecontributed by noise introduced into the gray-scale data appears to beproportional to the magnitude of change in the gray-scale value. Thus,the distorting effects of noise within symbols of a signal are notnecessarily uniform. A noise-induced change of a transmitted symbol to aclosely related, received symbol may produce far less distortion than anoise-induced change of a transmitted symbol to a very different,received symbol.

The DUDE models the non-uniform distortion effects of particular symboltransitions induced by noise with a matrix Λ. FIG. 6 displays one formof the symbol-transformation distortion matrix Λ. An element d_(a) _(i)_(→a) _(j) of the matrix Λ provides the relative distortion incurred bysubstituting the symbol “a_(j)” in the noisy or recovered signal for thesymbol “a_(i)” in the clean signal. An individual column j of the matrixΛ may be referred to as λ_(j) or λ_(a) _(j) .

FIG. 7 illustrates computation of the relative distortion, with respectto the clean signal, expected from replacing a symbol “a_(a)” in areceived, noisy signal by the symbol “a_(x)”. As shown in FIG. 7,element-by-element multiplication of the elements of the column vectorsλ_(a) _(x) and π_(a) _(a) , an operation known as the Shur product oftwo vectors, and designated in the current discussion by the symbol □,produces the column vector λ_(a) _(x) □π_(a) _(a) in which the i-thelement is the product of a distortion and probability, d_(a) _(i) _(→a)_(x) P_(a) _(i) _(→a) _(a) , reflective of the relative distortionexpected in the recovered signal by replacing the symbol a_(a) in thenoisy symbol by the symbol “a_(x)” when the symbol in the originallytransmitted, clean signal is “a_(i).”

FIG. 8 illustrates use of the column vector λ_(a) _(x) □π_(a) _(a) tocompute a distortion expected for replacing “a_(a)” in the metasymbolba_(a)c in a noisy signal s_(noisy) by the replacement symbol “a_(x)”.In the following expression, and in subsequent expressions, the vectorss_(noisy) and s_(clean) denote noisy and clean signals, respectively. Adifferent column vector q can be defined to represent the occurrencecounts for all symbols in the clean signal that appear at locations inthe clean signal that correspond to locations in the noisy signal aroundwhich a particular context [b, c] occurs. An element of the columnvector q is defined as:q(s _(noisy) ,s _(clean) ,b,c)[a _(a) ]=|{i:s _(clean) [i ]=a _(a),(s_(noisy) [i−k],s _(noisy) [i−k+1], . . . , s _(noisy) [i−1])=b, (s_(noisy) [i+1], s _(noisy) [i+2], . . . , s _(noisy) [i+k])=c}|,where s_(clean)[i] and s_(noisy)[i] denote the symbols at location i inthe clean and noisy signals, respectively; and

-   -   a_(a) is a symbol in the alphabet A.        The column vector q(s_(noisy),s_(clean),b,c) includes n elements        with indices a_(a) from “a₁” to “a_(n),” where n is the size of        the symbol alphabet A. Note that the column vector        q(s_(noisy),s_(clean),b,c) is, in general, not obtainable,        because the clean signal, upon which the definition depends, is        unavailable. Multiplication of the transpose of the column        vector q(s_(noisy),s_(clean),b,c),        q^(T)(s_(noisy),s_(clean),b,c), by the column vector λ_(a) _(x)        □π_(a) _(a) produces the sum of the expected distortions in the        column vector times the occurrence counts in the row vector that        together provide a total expected distortion for replacing        “a_(a)” in the metasymbol ba_(a)c in s_(noisy) by “a_(x)”. For        example, the first term in the sum is produced by multiplication        of the first elements in the row vector by the first element in        the column vector, resulting in the first term in the sum being        equal to q^(T)(s_(noisy),s_(clean),b,c)[a₁](P_(a) _(1 →) _(a)        _(a) d_(a) _(1 →) _(a) _(x) ) or, in other words, a contribution        to the total distortion expected for replacing “a_(a)” by        “a_(x)” in all occurrences of ba_(a)c in s_(noisy) when the        corresponding symbol in s_(clean) is a₁. The full sum gives the        full expected distortion:

$\begin{matrix}{{{{q^{T}\left( {s_{noisy},s_{clean},b,c} \right)}\left\lbrack a_{1} \right\rbrack}\left( {p_{a_{1->}a_{\alpha}}d_{a_{1->}a_{x}}} \right)} +} \\{{{{q^{T}\left( {s_{noisy},s_{clean},b,c} \right)}\left\lbrack a_{2} \right\rbrack}\left( {p_{a_{2->}a_{\alpha}}d_{a_{2->}a_{x}}} \right)} +} \\{{{{q^{T}\left( {s_{noisy},s_{clean},b,c} \right)}\left\lbrack a_{3} \right\rbrack}\left( {p_{a_{3->}a_{\alpha}}d_{a_{3->}a_{x}}} \right)} +} \\\vdots \\{{{q^{T}\left( {s_{noisy},s_{clean},b,c} \right)}\left\lbrack a_{n} \right\rbrack}\left( {p_{a_{n->}a_{\alpha}}d_{a_{n->}a_{x}}} \right)}\end{matrix}$

As discussed above, DUDE does not have the advantage of knowing theparticular clean signal, transmitted through the noise-introducingchannel that produced the received noisy signal. Therefore, DUDEestimates the occurrence counts, q^(T)(s_(noisy),s_(clean),b,c), ofsymbols in the originally transmitted, clean signal, by multiplying therow vector m^(T)(s_(noisy),b,c) by Π⁻¹ from the right. FIG. 9 showsestimation of the counts of the occurrences of symbols “a₁”-“a_(n)” forthe clean signal.

The resulting expressionm^(T)(s_(noisy),b,c)Π⁻¹(λ_(a) _(x) □π_(a) _(a) )obtained by substituting m^(T)(s_(noisy),b,c) Π⁻¹ forq^(T)(s_(noisy),s_(clean),b,c) represents DUDE's estimation of thedistortion, with respect to the originally transmitted clean signal,produced by substituting “a_(x)” for the symbol “a_(a)” within thecontext [b, c] in the noisy signal s_(noisy). DUDE denoises the noisysignal by replacing “a_(a)” in each occurrence of the metasymbol ba_(a)cby that symbol “a_(x)” providing the least estimated distortion of therecovered signal with respect to the originally transmitted, cleansignal, using the above expression. In other words, for each metasymbolba_(a)c, DUDE employs the following transfer function to determine howto replace the central symbol a_(a):

${g_{a}^{k}\left( {b,a_{\alpha},c} \right)} = {\frac{\arg\mspace{14mu}\min}{a_{x} = {a_{1}\mspace{14mu}{to}\mspace{14mu} a_{n}}}\left\lbrack {{m^{T}\left( {s_{noisy},b,c} \right)}{\prod^{- 1}\left( {\lambda_{a_{x}}\bullet\mspace{14mu}\pi_{a_{\alpha}}} \right)}} \right\rbrack}$In some cases, the minimum distortion is produced by no substitution or,in other words, by the substitution a_(x) equal to a_(a).

FIG. 10 illustrates the process by which DUDE denoises a noisy, receivedsignal. First, as discussed above, DUDE compiles counts for all or aportion of the possible metasymbols comprising each possible symbol“a_(i)” within each possible context [b, c]. As discussed above, thecounts are stored in column vectors m(s_(noisy),b,c). In the next pass,DUDE again passes a sliding window over the noisy signal 1002. For eachmetasymbol, such as metasymbol 1004, DUDE determines the relativedistortions of the recovered signal with respect to the clean signalthat would be produced by substituting for the central character of themetasymbol “a_(a)” each possible replacement symbol “a_(i)” in the rangei=1 to n. These relative distortions are shown in table 1006 in FIG. 10for the metasymbol 104 detected in the noisy signal 1002. Examining therelative distortion table 1006, DUDE selects the replacement symbol withthe lowest relative distortion, or, in the case that two or more symbolsproduce the same relative distortions, selects the first of the multiplereplacement symbols with the lowest estimated distortion. In the exampleshown in FIG. 10, that symbol is “a₃” 1008. DUDE then replaces thecentral symbol “a_(a)” 1010 in the noisy signal with the selectedreplacement symbol “a₃” 1012 in the recovered signal 1014. Note that therecovered signal is generated from independent considerations of eachtype of metasymbol in the noisy signal, so that the replacement symbolselected in a previous step does not affect the choice for a replacementsymbol in a next step for a different metasymbol. In other words, thereplacement signal is generated in parallel, rather than substitution ofsymbols directly into the noisy signal. As with any general method, theabove-described method by which DUDE denoises a noisy signal can beimplemented using various data structures, indexing techniques, andalgorithms to produce a denoising method that has both linear time andlinear working-data-set complexities or, in other words, the timecomplexity is related to n, the length of the received, noisy signal, bymultiplication by a constant, as is the working-data-set complexity.

The examples employed in the above discussion of DUDE are primarily1-dimensional signals. However, as also discussed above, 2-dimensionaland multi-dimensional signals may also be denoised by DUDE. In the2-and-multi-dimensional cases, rather than considering symbols within alinear context, symbols may be considered within a contextualneighborhood. The pixels adjacent to a currently considered pixel in a2-dimensional image may together comprise the contextual neighborhoodfor the currently considered symbol, or, equivalently, the values of acurrently considered pixel and adjacent pixels may together comprise a2-dimensional metasymbol. In a more general treatment, the expressionm^(T)(s_(noisy),b,c)Π⁻¹(λ_(a) _(x) □π_(a) _(a) ) may be replaced by themore general expression:m^(T)(s_(noisy),η)Π⁻¹(λ_(a) _(x) □π_(a) _(a) )where η denotes the values of a particular contextual neighborhood ofsymbols. The neighborhood may be arbitrarily defined according tovarious criteria, including proximity in time, proximity in display orrepresentation, or according to any arbitrary, computable metric, andmay have various different types of symmetry. For example, in theabove-discussed 1-dimensional-signal examples, symmetric contextscomprising an equal number of symbols k preceding and following acurrently considered symbol compose the neighborhood for the currentlyconsidered symbol, but, in other cases, a different number of precedingand following symbols may be used for the context, or symbols eitheronly preceding or following a current considered symbol may be used.

Lossless and Lossy Compression

As mentioned above, lossless compression involves compressing data or asignal stream in a way that allows the original data or signal stream tobe exactly restored, without distortion, upon decompression. One popularlossless data compression technique is known as the “Lempel-Ziv”compression technique. The Lempel-Ziv technique is most usuallydescribed using a simple example. FIGS. 11A-B illustrate Lempel-Zivlossless data compression. In FIG. 11A, an initial, binary data signal1102 is first shown. In a first step, the binary data signal is parsed,from left to right, to detect and tabulate each as-yet-not-tabulatedsubstring. For example, the first “0” symbol 1104 constitutes, byitself, a first, as-yet-not-tabulated substring. Two vertical lines1106-1107 are placed above and below, respectively, the parsed datasignal 1108 to indicate the first tabulated substring boundary. Then,parsing of the next as-yet-not-encountered tabulated substring isundertaken. The first symbol, “0” 1110, comprises a substring of length1, but this substring has already been recognized and tabulated.Therefore, parsing continues with the next symbol “1” 1112. The “0” 1110and “1” 1112 together constitute an as-yet-not-tabulated substring “01,”and so vertical lines 1114 and 1115 are placed above and below theparsed data signal 1108 to indicate the second tabulated substringboundary. The parsing continues in a left-to-right fashion, resulting inthe 13 as-yet-not-tabulated substrings with corresponding substringindices 1-13. In a second step, an encoded data signal 1116 is createdby replacing each parsed, tabulated substring in the parsed data signal1108 with a substring-index/symbol pair. Each substring-index/symbolpair, in the example in FIG. 11A, includes 5 bits that include a first,4-bit index and a second 1-bit symbol. Because each as-yet-not-tabulatedsubstring comprises an already-tabulated-substring prefix and a single,final binary symbol that confers as-yet-not-tabulated status to thesubstring, each parsed substring can be represented as the index of thealready-tabulated-substring prefix followed by the binary symbol thatconfers as-yet-not-tabulated status to the substring. For example, thefirst parsed substring “0” 1104 in the parsed data signal 1108 isrepresented by the substring-index/symbol pair including index 0 (0000in binary notation) indicating no prefix, and the single binary symbol“0” 1118 in the encoded data signal 1116. Similarly, the next parsedsubstring comprising symbols “0” 1110 and “1” 1112 is represented by thesubstring-index/symbol pair 1120 including index 1 (0001 in binarynotation) and the binary symbol “1.” In this simple embodiment of theLempel-Ziv method, the number of bits used to represent the index of thealready-encountered-substring prefix is no smaller than the logarithm tothe base 2 of the number of parsed substrings in the entire datasequence. This allows each parsed substring to have a unique index.

Implementations of the Lempel-Ziv technique normally construct acodebook, as shown in FIG. 11B, containingalready-tabulated-substring/substring-index-symbol pair entries. In theexample in FIGS. 11A-B, a fixed index length is employed, but in moresophisticated approaches, variable-length index fields within codewordsare employed. Decoding of an encoded data string is carried out inreverse fashion, along with construction of the codebook. In the Exampleshown in FIG. 11A, the encoded data signal 1116 is initially longer thanthe original data signal 1102, but, as encoding progresses, longer andlonger substrings are parsed and encoded, generally leading tocompression ratios (ratio of compressed size to uncompressed size) lessthan one and on many types of data, such as English text, substantiallyless than one. As discussed above, lossy compression techniques resultin loss of information or, in other words, decompressed data that isdistorted with respect to the original compressed data. However, lossycompression techniques may produce dramatically better compressionratios than those produced by lossless compression techniques.

FIG. 12 illustrates the encoding scheme for the vector quantizationlossy compression technique. The vector quantization technique involvesbreaking up a data stream for compression into a series of vectors offixed length. For each vector, a shortened, encoded vector issubstituted to produce the compressed version of the data. There is amany-to-one relationship between the vectors and encoded vectors. Indecompression, the encoded vectors are expanded back into full-lengthvectors. However, because of the many-to-one relationship betweenvectors and encoded vectors, expansion of encoded vectors to full-lengthvectors results in a distorted, less than accurate decompressed datawith respect to the original data.

FIG. 12 shows a vector-quantization encoding scheme for encoding vectorsof length 4 by encoded vectors of length 2. In FIG. 12, all possiblevectors of binary digits of length 4 are enumerated within the 16-entrytable 1202. The possible vectors are partitioned into four subsets: (1)a first subset containing only the vector “0000” 1204; (2) a secondpartition 1206 containing the three vectors “0001,” “0010,” and “0011”;(3) a third partition 1208 containing the three vectors “0100,” “1000,”and “1100”; and (4) a large, fourth partition 1210 including allremaining binary vectors of length 4. The encoded vectors of length 2corresponding to each of the four partitions are “00,” “01,” “10,” and“11,” respectively. The encoded vectors are simply the numbers 0, 1, 2,and 3 in base 2 notation. When encoded vectors are expanded, duringdecompression, the vectors of length 4 marked by ovals in each of thefour partitions are chosen for the expansion. For example, the encodedvector “01” is expanded to the vector of length 4 “0011” wherever itoccurs in the compressed data stream. Therefore, vectors “0001,” “0010,”and “0011” that occurred in the original data stream all occur as thevector “0011” in the decompressed data stream. Thus, compression anddecompression result in loss of information.

FIG. 13 illustrates lossy compression by the vector quantizationtechnique described with reference to FIG. 12, using the encoding tableillustrated in FIG. 12. The original data stream 1302 is broken up intovectors of length 4, and encoded as encoding vectors of length 2 toproduce the compressed data stream 1304. Decompression of the compresseddata stream results in the recovered data stream 1306. Those vectors oflength 4 in the recovered data stream marked by asterisks in FIG. 13have been altered by the compression/decompression operation. In vectorquantization, the compression ratio is exactly determined by therespective length of the vectors and encoding vectors, but the potentialdistortion in the decompressed data stream is inversely related to thecompression ratio. In other words, the greater the compression achieved,the greater potential distortion in the decompressed data stream.Arbitrary vector quantization encoding generally ignores the occurrencestatistics for various symbols and metasymbols within the uncompresseddata stream, potentially leading to significant distortion. In moreelaborate vector quantization techniques, encoding vectors are carefullychosen to provide the greatest chance of reproducing the original datastream, based on the statistics of occurrences of the fixed lengthvectors in the original uncompressed data. However, even these moreelaborate techniques produce significant distortion at sufficiently highdegrees of compression.

Compression Methods Using a Denoiser

Embodiments of the present invention provide compression of data byapplying a discrete denoiser, or other type of denoiser, to initiallyprocess the data, and then using either a lossless compressiontechnique, such as the Lempel-Ziv technique, or a lossy compressiontechnique to compress the output of the denoiser to produce the finalcompressed data. Decompression is carried out using the decompressiontechnique corresponding to the compression technique to recover theoutput from the denoiser. The overall compression technique is generallylossy, since information is lost when the original data is processed bythe denoiser, or denoised. In one embodiment, the combination of usingthe DUDE and the Lempel-Ziv technique represents the combination of twouniversally applicable techniques to carry out lossy compression,without the need for particular knowledge of the data to be compressed.As discussed above, effective lossy compression needs careful analysisof the data to be compressed in order to avoid introducing readilyperceptible discontinuities and noise. The lossy compression techniquesof the present invention, by contrast, can be carried out on a datasignal without such analysis, because the DUDE tends generally to filterhigh-entropy noise. In alternative embodiments of the present invention,any of a number of different compression techniques may be employed in asecond step of a general two-step process, in which the first stepinvolved denoising. The compression method chosen for the second stepdepends on the nature of the data to be compressed, variouscomputational and economic efficiencies related to the compressionproblem, and other considerations. For example, the JPEG-LS compressionmethod is one of many compression techniques suited for compression ofimages.

FIG. 14 is a control-flow diagram for a general embodiment of thepresent invention. In a first step 1402, the data to be compressed isreceived. In a next step 1404, the received data is denoised using adenoising method such as the discrete denoiser described in a previoussubsection. The denoised data is then compressed, in a third step 1406,using a compression technique, such as the lossless Lempel-Zivtechnique, to produce a final, compressed data. Denoising the originaldata filters the high entropy portions of the data, greatly facilitatinglossless compressing and generally leading to better compression ratios.

In more elaborate embodiments, it may be desirable to tune the denoisingstep iteratively prior to the compression step, in order to produce themost desirable denoised data. FIG. 15 illustrates a self-tuningimplementation of the compression method of various embodiments of thepresent invention. In step 1502, the data to be compressed is received.In step 1504, a denoising method is used to denoise the received data.In step 1506, a distortion metric is computed in order to ascertain thelevel of distortion introduced by the denoising step. If this level ofdistortion is acceptable, as determined in step 1508, then compressionis undertaken in step 1510. Otherwise, new denoising parameters areselected, in step 1512, and control returns to step 1504 for asubsequent iteration of denoising. If the distortion level isacceptable, compression is carried out in step 1510. The resultingcompression ratio, or compression rate, is determined in step 1514 bycomparing the size of the compressed data to the size of the originaldata. If the compression rate is acceptable, then compression iscomplete. Otherwise, control flows back to step 1512, where newdenoising parameters are selected in order to again denoise the inputdata in a way to provide a better compression ratio in the compressionstep 1510. Note that, in general, some limit on the number of iterationsof denoising and compression may be set so that the overall compressiontechnique is carried out within some maximum amount of time or followingsome maximum expenditure of computing resources.

The tuning of the denoising parameters, in step 1512, may beaccomplished in many different ways. For example, tuning may be a purelyautomated process, a semi-automated process, or may be carried outmanually, by a user who interacts with a user interface featuringknob-like interfaces that allow the user to adjust the denoisingparameters. Any of the denoising parameters, including the contexts thatare considered, the length of the contexts k, the channel transitionmatrix Π, and a distortion matrix Λ can be adjusted. In one method, thedenoiser is tuned by modifying the channel transition matrix Π to findan optimal channel transition matrix Π that produces an optimizeddenoised signal leading to greatest compressibility and acceptabledistortion. The optimization may be facilitated by heuristic approaches.One heuristic is to set the entries of each row i of Π, denoted π_(i,1),. . . , π_(i,n) to be the probability distribution having the largestentropy subject to the constraint that

${\sum\limits_{j}{\pi_{i,j}\lambda_{i,j}}} < \Delta$where λ_(i,j) are the entries of the ith row of the loss/distortionmatrix Λ. A good initial value for Δ is the desired distortion levelbetween the lossy representation and the original data. In this case,the above-described iterative approach may adjust Δ in order to convergeon a desired ratio between compression efficiency and signal distortion.

Although the present invention has been described in terms of aparticular embodiment, it is not intended that the invention be limitedto this embodiment. Modifications within the spirit of the inventionwill be apparent to those skilled in the art. For example, although adiscrete denoiser, such as the denoiser described in a previoussubsection, is preferred, any of many different types of denoisers maybe used in the lossy compression schemes that represent embodiments ofthe present invention. Similarly, many different compression techniquesmay be used to compress the denoised signal. As discussed above, any ofan almost limitless number of methods for tuning the denoiser parametersmay be used in internal feedback controls within the lossy compressiontechnique in order to produce optimal compression levels for specifiedmaximum levels of distortion. The lossy compression technique may beimplemented as a software program, as a higher level script program thatcalls various existing denoiser and lossy compression modules, or may beeven implemented in hardware or a combination of hardware and softwarewithin communications and data storage devices.

The foregoing description, for purposes of explanation, used specificnomenclature to provide a thorough understanding of the invention.However, it will be apparent to one skilled in the art that the specificdetails are not required in order to practice the invention. Theforegoing descriptions of specific embodiments of the present inventionare presented for purpose of illustration and description. They are notintended to be exhaustive or to limit the invention to the precise formsdisclosed. Obviously many modifications and variations are possible inview of the above teachings. The embodiments are shown and described inorder to best explain the principles of the invention and its practicalapplications, to thereby enable others skilled in the art to bestutilize the invention and various embodiments with various modificationsas are suited to the particular use contemplated. It is intended thatthe scope of the invention be defined by the following claims and theirequivalents:

1. A method for compressing data comprising: receiving the data;repeatedly denoising the data using a denoiser to produce denoised data,each time selecting different denoising parameters wherein denoisingparameters comprise symbol compositions of contexts within which symbolsare evaluated by the denoiser; lengths of the contexts within whichsymbols are evaluated by the denoiser; values stored in achannel-transition matrix Π; and values stored in a distortion matrix Λ,until a distortion computed for the denoised data is lower than athreshold distortion; and compressing the denoised data to producecompressed data.
 2. The method of claim 1 wherein denoising andcompressing are repeated until compressed data with a size lower than athreshold size is produced.
 3. The method of claim 1 wherein repeatedlydenoising the data using a denoiser to produce denoised data, each timeselecting different denoising parameters, until a distortion computedfor the denoised data is lower than a threshold distortion is carriedout by one of: an automated system; a semi-automated system thatincludes user interaction; and a manually controlled system in which auser interactively selects and adjusts denoising parameters.
 4. Themethod of claim 1 wherein the denoiser is tuned by modifying thechannel-transition matrix Π to find an optimal channel transition matrixΠ that produces an optimized denoised signal leading to greatestcompressibility and acceptable distortion.
 5. The method of claim 4further including: setting entries of each row i of thechannel-transition matrix Π, denoted π_(i,1), . . . , π_(i,n) to be aprobability distribution having a largest entropy subject to aconstraint that ${\sum\limits_{j}{\pi_{i,j}\lambda_{i,j}}} < \Delta$ where λ_(i,j) are entries of the ith row of the distortion matrix Λ andΔ is a threshold distortion level.
 6. The method of claim 5 wherein aninitial value for Δ is selected as a desired distortion level betweenlossy representation of the data and the uncompressed data.
 7. Themethod of claim 5 wherein Δ is iteratively adjusted as a denoisingparameter in order to converge on a desired ratio between compressionefficiency and signal distortion.
 8. The method of claim 1 whereincompressing the data further includes compressing the data using one of:a lossy decompression method; and a lossless decompression method. 9.The method of claim 1 further including subsequently decompressing thecompressed data by reversing the compression using one of: a lossydecompression method; and a lossless decompression method.
 10. Acomputer readable storage medium containing executable instructionswhich, when executed in a processing system, causes the system toperform a method for compressing data comprising: receiving the data;repeatedly denoising the data using a denoiser to produce denoised data,each time selecting different denoising parameters wherein denoisingparameters comprise symbol compositions of contexts within which symbolsare evaluated by the denoiser; lengths of the contexts within whichsymbols arc evaluated by the denoiser; values stored in achannel-transition matrix Π; and values stored in a distortion matrix Λ,until a distortion computed for the denoised data is lower than athreshold distortion; and compressing the denoised data to producecompressed data.
 11. The computer readable storage medium of claim 10wherein denoising and compressing are repeated until compressed datawith a size lower than a threshold size is produced.
 12. The computerreadable storage medium of claim 10 wherein repeatedly denoising thedata using a denoiser to produce denoised data, each time selectingdifferent denoising parameters, until a distortion computed for thedenoised data is lower than a threshold distortion is carried out by oneof: an automated system; a semi-automated system that includes userinteraction; and a manually controlled system in which a userinteractively selects and adjusts denoising parameters.
 13. The computerreadable storage medium of claim 10 wherein the denoiser is tuned bymodifying the channel-transition matrix Π to find an optimal channeltransition matrix Π that produces an optimized denoised signal leadingto greatest compressibility and acceptable distortion.
 14. The computerreadable storage medium of claim 13 wherein the method further includes:setting entries of each row i of the channel-transition matrix Π,denoted π_(i,1), . . . , π_(i,n) to be a probability distribution havinga largest entropy subject to a constraint that${\sum\limits_{j}\;{\pi_{i,j}\lambda_{i,j}}} < \Delta$  where λ_(i,j)are entries of the ith row of the distortion matrix Λ and Δ is athreshold distortion level.
 15. The computer readable storage medium ofclaim 14 wherein an initial value for Δ is selected as a desireddistortion level between lossy representation of the data and theuncompressed data.
 16. The computer readable storage medium of claim 14wherein Δ is iteratively adjusted as a denoising parameter in order toconverge on a desired ratio between compression efficiency and signaldistortion.
 17. The computer readable storage medium of claim 10 whereincompressing the data further includes compressing the data using one of:a lossy decompression method; and a lossless decompression method. 18.The computer readable storage medium of claim 10 wherein the methodfurther includes subsequently decompressing the compressed data byreversing the compression using one of: a lossy decompression method;and a lossless decompression method.